If $\int \frac{1-x^7}{x\left(1+x^7\right)} d x=a \ln |x|+b \ln \left|x^7+1\right|+C$, then

$a=1, b=-\frac{2}{7}$

We have,

$\int \frac{1-x^7}{x\left(1+x^7\right)} d x=a \ln |x|+b \ln \left|x^7+1\right|+C$

Differentiating both sides w.r. to, $x$, we get

$\frac{1-x^7}{x\left(1+x^7\right)}=\frac{a}{x}+7 b \frac{x^6}{x^7+1}$

$\Rightarrow 1-x^7=a\left(1+x^7\right)+7 b x^7 \Rightarrow a=1, b=-\frac{2}{7}$